Transverse and longitudinal momentum projectors ($P_L{}^{\mu }{}_{\nu }=k^{\mu }{k}_{\nu }/k^2,P_T=I-P_L$)

At non-null four-momentum, these complementary projectors satisfy $P_L^2=P_L$, $P_T^2=P_T$ and $P_L{P}_T=0$. The gauge-fixed Maxwell kinetic operator is $-k^2(P_T+{\xi }^{-1}{P}_L)$; its inverse gives propagator $-i(P_T+\xi P_L)/k^2$. Transversality here removes one four-dimensional longitudinal direction, leaving rank three; it does not by itself project onto the two physical photon polarizations. The pole requires the Feynman i-epsilon prescription.